**Functional Analysis 2011 (SF 2707)**

Ari Laptev

Course information

OBS! The FA exam will be on Friday, June 10, in room 3721 between 10:00 - 14:00.

The main goal is to give an introduction to the basics of functional analysis and operator theory, and to some of their (very numerous) applications.

First lecture will be on Tuesday **January 18 between 14:15-16:00 ** in the seminar room
** 3721**, institutionen för matematik.

We continue our lectures every second ** Tuesday between 13:15-16:00 in 3721 with one exception: our second lecture will be on January 31 between 10:15-13:00 in room E3.**

Content

- Banach spaces. Bounded operators.
- The Banach-Steinhaus theorem.
- The open mapping theorem.
- The closed graph theorem.
- Dual spaces. The Hahn-Banach theorem.
- Dual spaces to L
^{p}and C(X). - Weak topologies. The Banach-Alaoglu theorem.
- Adjoint operators.
- Compact operators and their spectrum.
- Fredholm alternative.
- Hilbert spaces and operators on Hilbert spaces.
- The spectral theorem for self-adjoint operators on Hilbert spaces.
- Fredholm determinants.
- Unbounded operators.

Lectures and home excercises - week 2

Lectures and home excercises - week 3

Lectures and home excercises - week 4

Lectures and home excercises - week 5

Lectures and home excercises - week 6

Lectures and home excercises - week 7

Lectures and home excercises - week 8

Lectures and home excercises - week 9

**Books**

**Avner Friedman, **
* Foundations of Modern Analysis, *
Dover Publications, Inc., 1982.

**Other books**

**M. Reed and B. Simon,**
* Methods of modern mathematical physics. I. Functional Analysis, *
Academic Press, 1980 (First part of a multi-volume textbook on mathematical physics.
It gives a firm functional analysis background for use in the mathematical
physics).

**M.S. Birman and M.Z. Solomyak, **
* Spectral Theory of Self-Adjoint Operators in Hilbert Space, *
D.Reidel Publishing Company, 1987.

**P. Lax, **
* Functional Analysis, *
John Wiley & Sons, 2002 (Includes sections on the spectral resolution and
spectral representation of self adjoint operators, invariant subspaces,
strongly continuous one-parameter semigroups, the index of operators,
the trace formula of Lidskii, the Fredholm determinant, and more).